J. Biomehnzics,
Vol. 3.p.
495-5 I I.
Pergamon Press. 1970.
MECHANICAL
Printed in Great Britain
PROPERTIES
OF CRANIAL
BONE*
JAMES H. McELHANEY,
JOHN L. F0GLE.t
JOHN W. MELVIN,
RUSSELL R. HAYNES,? VERNE L. ROBERTS and NABIH M. ALEMt
Highway Safety Research Institute, The University of Michigan, Ann Arbor, Mich. 48105.
U.S.A.
Abstract-Samples
of human and Macaca mularta cranial bone have been tested quasistatically
in tension, compression, simple shear, and torsion. The results of these experiments have been
analyzed, taking into account observed anisotrophies and varying structures. Statistical correlations of properties have been made with density and a model proposed that summarizes these
results. The cranial bones appear to be transversely isotropic and they are generally much
stronger and stiffer in the transverse or tangent to the skull direction in comparison to the radial
direction. The structure of the dip& region was found to be highly variable and this strongly
influenced many of the mechanical responses. The model, however. explains much of the observed variation.
INTRODUCTION
cause of death and disability in this country
(Subcommittee
on Head Injury, 1969), with
an estimate that head injuries occur in 71 per
cent of persons injured in motorcycle accidents and in 70 per cent of persons injured
in accidental falls in the home. It is further
estimated that in 1967 there were 112,000
fatalities and 400,000 permanent impairments
due to accidental causes. While data is not
available as to what proportion of these were
head injuries, there is little doubt that the
percentage is discouragingly high.
The literature on the mechanical properties
of the skull is quite modest. Evans and Lissner
(1957) have measured the tensile and compressive strength of embalmed human parietal
bone but they have not measured strains
during the test and therefore could not report
the modulus of elasticity. Dempster ( 1967)
has made a detailed study of the cortical grain
structure of the human skull using the split
line method. He observed a random orientation of grain structure for the inner and outer
table of the brain case, supporting our observation of isotropy in the tangential direction
for skull bone. Wood (1969) recently finished
a detailed study of the tensile properties of
the advent of high speed air and land
transportation,
engineers have become increasingly aware of the mechanical frangibility
of the human body. Thus, we have seen the
evolution of various isolating and load distributing devices ranging from seat belts and
padded sun visors to ejection seats, crash
helmets and acceleration couches. While there
is much information available regarding the
response of inanimate material to vibration
and impact, there is an equal dearth of knowledge pertaining to the mechanical properties
of biological materials. Therefore, the design
of much support equipment is often based on
intuition because of this lack of information.
Knowledge of this type is valuable in the treatment of injuries since it serves to identify the
mechanism of trauma. A rational therapy for
head injury, for example, cannot be developed
until a quantitative description of the mechanical properties of the tissues of the head is
obtained.
The National Safety Council has shown
that accidents are the fourth leading cause of
death surpassed only by heart disease, cancer
and stroke. Head injury ranks as a major
WITH
*Received
3 1 March
tBiomechanics
1970.
Laboratory, West Virginia University, Morgantown, W.V. 26505, U.S.A.
495
496
J. H. McELHANEY
miniature specimens cut from the compact
bone of the inner and outer tables while Melvin
et al. (1969) has presented
a preliminary
report on the mechanical behavior of the diploe layer in compression.
The aim of the research described in this
paper is to specify the mechanical properties
of the skull relevant to the biomechanics of
head injury. This work is part of a program of
head injury research investigating the mechanical properties of all the tissues of the head.
The ultimate goal of this program is to determine the characteristics of these materials in
sufficient detail so that appropriate physical
and mathematical models of the head may
be constructed. These models will allow the
systematic study of various head injury mechanisms.
The skull is a complex structure made up of
several bones each with its own unique internal and external geometry. Mechanical
properties may be classified in three types:
geometrical
or spatial properties;
material
properties;
and structural properties.
The
measurement of all types of mechanical properties will be discussed in this paper. Each
category presents its own peculiar measurement problems and it is important to separate
these measurements as to type. In particular,
material properties should be separated from
structural
properties.
For instance,
load
deflection curves may be easily measured on
a variety of structures and are in general as
much property of the structure as of the
material of which the structure is composed.
It is only under certain highly restrictive conditions that such curves may be converted to
stress-strain curves or material properties that
are independent of the structural geometry.
The stress and strain distributions must be
known. It is possible to measure material
properties in tension or compression tests by
assuming and insuring that the stress and
strain are uniformly distributed. This can only
be completely true if the material under test is
homogeneous. A beam or simple shear test
requires a knowledge of the materials’ stress-
er al.
strain relations for interpretation and therefore yields only composite structural-material
properties that cannot be uncoupled without
additional information
about the material
(Gurdjian and Lissner, 1947). In the case of
the small samples of skull bone it is convenient
to consider the specimen homogeneous because the internal architecture
cannot be
determined in sufficient detail to allow a geometrical or structural analysis. Thus, the average stress and strain may be computed from
the load, deformation, and the gross dimensions of the specimen. But the average value
may be far from the maximum value because
of the non-homogeneous
microstructure
(Evans and Bang, 1967). It is beyond the
scope of this paper to deal with the influence
of various levels of organization of bone on
the stress distribution. Therefore,
average
values of stress and strain are reported
throughout. However, the model presented
here provides some insight into these problems
and hopefully will allow the use of these
average values in a more general way.
SPECIMEN PREPARATION AND LOCATION
The human skull bone used in this research
was obtained from three sources: embalmed
cadavers,
craniotomies
and autopsy. The
primary source was embalmed cadavers from
which 17 entire calvariums have been obtained with ages ranging from 56 to 73 yr at
death. Fresh material from craniotomies and
autopsy was also tested (40 donors) to verify
the results of the tests on embalmed material.
Previous work (McElhaney
et al.,
1964)
indicates that the mechanical properties of
embalmed bone are not significantly different
from immediate postmortem properties.
A fine grid reference system was used to
identify the location of test specimens from
the human and monkey skull. The system
used a 1 cm grid referenced from the coronal,
sagittal and lambdoidal sutures and originating from the bregma and lambda points. The
system is based entirely on the landmarks on
the top of the skull and has been used in an
Fig. I Grid reference
system
MECHANICAL
PROPERTIES
attempt to correlate the mechanical properties
with position (Fig. 1).
Three types of specimens were prepared
,from human skull bone. Specimen type ‘A’
was a lo-mm dia. cylinder prepared by coring
the skull with a Stryker oscillating core saw.
Water was applied to the saw during cutting
to prevent excessive heating. The thickness
‘r’ was the skull thickness. These specimens
are used in simple shear and torsion tests.
Specimen type ‘B’ was prepared from a lomm core. Specially designed fixtures with
micrometer
drives were used to precisely
grind a cuboidal shape with parallel surfaces.
All machining was done wet to avoid excessive
heating of the specimen. These specimens
were used in the triaxial compression tests.
Specimen type ‘C’ was a reduced section
tension specimen produced by grinding a
reasonably flat slice of bone between steel
templates. Because of the parallel side requirement only a few specimens of this type could
be made from each skull.
Only one type of monkey specimen had
been used because of the extreme thickness
of the monkey skull. This specimen was a
curved flap cut from the skull and ground flat
and parallel on the edges. It was used for compression tests in the tangential to the skull
directions. Figure 2 shows the shape and
dimensions of these specimens. After preparation, the specimens were kept damp
with isotonic saline buffered with calcium. All
tests were made on bone in a moist condition.
OF CRANIAL
Tension
497
BONE
Monkey Specimen
Specimen
Fig. 2. Specimen types and sizes.
RiRLt
,
7;
STRUCTURE AND GEOMETRY
The skull bones considered here were the
frontal, right and left parietal, and the occipital. Figure 3 compares a section through the
right and left parietal of the human with the
Rhesus monkey (Macaca mulutfa). In the
human, these bones show a well-defined shell
of compact bone separated by a core of spongy
cancellous
bone (diploe). Compact
bone
surrounds and reinforces the sutures. The
thickness of the spongy core increases toward
the center of the bone away from the sutures.
left
Fig. 3. Sections
through human and primate (Mocacu
bone.
md~tta)parietal
The spongy bone is quite variable in structure
with marrow spaces normally ranging from 3mm dia. down to microscopic size. The outer
J. H. McELHANEY
498
table of compact
bone is quite smooth.
The inner surface presents depressions corresponding to the cerebral convolutions,
and
numerous furrows for the ramifications of
various blood vessels. The structure of these
bones in the Rhesus monkey is quite different
from the human for there is much less diplo
and where it occurs it is generally one to two
pores thick.
Due to the sandwich construction of the
skull bone, the test samples previously described have a variable structural geometry.
These specimens contain the full cross section
except they are ground flat on the top and
bottom corresponding to the inside and outside of the skull. This results in a specimen
with an inner and outer table of compact bone
and a core of cancellous bone. The mechanical
properties of compact bone are much more
uniform and predictable than cancellous bone.
The marrow spaces and trabecular arrangement of the diploe are highly variable, this
giving rise to a wide range of mechanical
responses.
In order to ascribe meaningful
material properties to this cancellous bone,
one must have a correlation with the internal
geometry. We are currently performing analysis of a variety of models to develop this
correlation
but with limited success. The
limitation is the amount of geometrical detail
that is practical to build into the model.
%XPERIMJWTAL
et al.
type ‘B’ specimens of human bone and type
‘C’ specimens of monkey bone (McElhaney
and Byars, 1966). The type ‘B’ specimens
are loaded first in one direction to about 4
the ultimate strength, while the deformation
is measured in the direction of the load and
also perpendicular to it. This is repeated in
the other two directions yielding the stiffness
and Poisson’s ratio for the three axes, that is
the radial and the two tangential to the head
directions. The load is taken to failure in the
third of these tests. Due to the thinness of
monkey bone the type ‘C’ specimens are
loaded in the tangential direction only.
The load was applied by a Tinius Olsen
electromatic Testing Machine at a constant
velocity of O-01 in./min. The load was monitored by a very stiff strain gage type load ring
while the deformation in the direction of the
load and perpendicular to the load were measured with very complient strain gaged cantilever contact arms (Fig. 4). These signals were
continuously recorded on an x-y, - yz recorder. The slope of these curves was used to
compute the modulus and Poisson’s ratio.
PROCEDURES
Density
The density of the specimens was determined by measuring the external dimensions
of the specimen to determine the volume and
by dividing this into the dry weight. After the
mechanical properties tests are completed, the
specimens were dried by baking at 105” until
no further weight loss is observed. This density was therefore the weight of dry bone per
unit volume and provides a convenient method
of estimating the porosity.
Compression
Compression
tests were performed
on the
Fig. 4. Compression test apparatus.
MECHANICAL
PROPERTIES
The ultimate strength was the stress corresponding to the maximum load the specimen
could withstand, generally characterized
by
a slope reversal. The ultimate strain was
taken as the magnitude of the strain at the
maximum load. The energy absorption capacity was determined as the work done on the
material per unit volume in loading to the
maximum load, and was proportional to the
area under the load-deflection diagram.
OF CRANIAL
499
BONE
‘\
,_
‘/
--‘>
--.
-LOADCELL
GUIDE
BALL
DRIVING
LOADER
8 SUPPORT
BASE
HEAD
DEFLECTOMETER
COLLET
IiOLDER
Tension
Tension tests to failure were performed on
miniature reduced section specimens. In some
cases the total cross-section was tested, while
in other cases the inner or outer table of compact bone was isolated and tested (see Wood
1969), for a complete description of this test).
The load was applied by the above mentioned
testing machine through specially designed
grips supported
on spherical bearings to
minimize eccentricity.
The load was monitored by a strain gage type load ring and the
deformation
was measured by a specially
designed clip gage with at-in. gage length or by
strain gages cemented to the bone surface.
The load deformation data was recorded on
an x-y
plotter and appropriate properties
determined as in the compression test.
Simple shear
A simple shear test was performed on type
‘A’ specimens by gripping the ends containing
the inner and outer table in collets and moving
one collet with respect to the other. A rigid
guide was used to maintain the axes of the
collets parallel (Fig. 5). Load and deformation were measured and recorded as in the
compression test. With this arrangement, the
diploe was loaded in simple shear. The collets
were fixed &in. apart resulting in a length to
diameter ratio less than one. In addition, the
collet assembly was supported in an indexing
cylinder which allowed rotation as a unit of
both cylinders and the specimens. Then the
shear load could be applied in various directions. Each specimen was loaded to approxi-
FIXED
HEAD
‘\- __
Fig. 5. Simple shear apparatus.
mately 3 the ultimate strength and the loaddeflection curve recorded. The specimen was
indexed 15” and loaded again. This procedure
was continued until the specimen had been
rotated 180” and the load axis then corresponded to the original one. Half the specimens were then rotated to the direction with
the steepest slope and loaded to failure and
the other half of the specimens were rotated
to the direction with the lowest slope and
loaded to failure. There was no significant
difference in the ultimate strength in simple
shear as determined from these two groups.
The ratio of the largest to the smallest slope
was a measure of isotropy for the specimen.
For example, an aluminum specimen composed of 5 flat leaves was tested. When loaded
parallel to the leaves, the slope of the loaddeflection curve was 16 times larger than the
slope when the load was applied perpendicular
to the leaves. Since the surface tractions in
this type of loading were unknown, the loaddeflection diagrams could not be converted to
stress or strain curves. By varying the load
500
directions in the
relative index of
This index is the
the smallest slope
specimen.
J. H. McELHANEY
manner described above, a
isotropy may be obtained.
ratio of the largest slope to
as obtained for a particular
Torsion
Collets similar to those described above
were used to grip and twist type ‘A’ specimens. Specially designed torque and angle of
twist transducers
continuously
monitored
these quantities and an on-line x-y plotter
drew the torque-angle of twist curve. This
curve was normalized to the shear stressstrain curve using the classical linear elastic
homogeneous
assumptions.
The specimen
was gripped in collets at both ends and torque
applied about the cylinder axis. The inner and
outer tables were constrained in collets and
since they were much stiffer and stronger than
the diploe this was essentially a test of the
diplot with negligible influence from the compact bone of the tables.
In this test, as well as the other tests previously described, the assumption of classical
elastic stress distributions was expedient but
certainly not justified in the small. The short
specimen
lengths and obvious structural
irregularities resulted in unknown stress variations. However, the assumed stress distributions were average values consistent with
the statics of the loading situation. The material properties
reported here are therefore
reasonably representative of the bulk properties of skull bone.
Hardness
A microhardness test has been developed
that allows the hardness of small regions
(0.005 in. dia.) to be measured (Zeniya et al.,
1964). This test was performed on Tukon
Microhardness Tester using a diamond pyramid indenter. The diameter of penetration
under a 100/g load was converted to Vickers
Hardness numbers. Four tests on the inner
table surface and four tests on the outer table
surface were made on each specimen.
et al.
RESULTS
A summary of the results of these tests is
presented in Tables 1 and 2. The high values
of the standard deviations are due to the
naturally occurring variations of the diploe
thickness and density. For human skull bone,
no significant difference was found for the
modulus and ultimate strength due to loading
in various tangential directions. Histological
studies of tangent sections of the inner and
outer tables and the diploe, revealed random
patterns without discernable geometrical organization. The index of isotropy, which is a
quite sensitive measure of isotropy, was low
and the maximum and minimum values of the
modulus upon which it was based occurred in
random directions. Thus, it was concluded
that skull bone is reasonably isotropic in directions tangent to the skull surface.
There was no significant difference (90 per
cent confidence level) between the microhardness tests for the inner and outer skull
surfaces. In addition, there was no significant
difference
(90 per cent confidence
level)
between the microhardness
tests for the
human and monkey inner and outer tables.
Since microhardness is a relative measure of
combined strength-modulus properties, it may
be concluded that there is little difference in
these properties for the inner and outer surfaces of the skull. This conclusion agrees with
the results of the tension tests on bone specimens from the inner or outer tables loaded in
the tangent to the skull direction. No significant difference was observed for the tensile
properties of bone from the inner and outer
tables.
The fourth column of Tables 1 and 2 indicates
the results of a student’s t test comparing the
mean value of the property for one skull with
the mean value of all the tests. This was done
in an attempt to determine whether the variation of the property within a single skull overshadowed the skull to skull variation. The results indicate that those properties related to
diploe structure vary significantly skull to
skull while those properties related more to
material behavior do not.
MECHANICAL
PROPERTIES
OF CRANIAL
50
BONE
Table 1. Properties of human cranial bone
Property
Skull thickness
in.
Diplot thickness
in.
Dry weight
density #/in3
Modulus compression
radial psi X 10’
Secondary modulus
compression radial
psi X 10”
Modulus compression
tangential direction
psi X lo5
Poisson’s ratio
compression radial
Poisson’s ratio
compression tangential
Ultimate strength
compression radial
psi X 10:’
Ultimate strength
compression tangential
psi X lo”
Ultimate strain
compression radial
in/in X lo+
Ultimate strain
compression tangential
in/in X 10e3
Energy absorption
compression radii
in #/in3
Energy absorption
compression tangential
in #/in”
Microhardness Vickers DPH
inner table
Microhardness Vickers DPH
outer table
Ultimate strength
diploe direct shear
psi X IO:%
Index of isotropy
Ultimate strength
diplot torsion
psi X lo”
Modulus torsion
diploe psi X 1OS
Ultimate strength
tension tangential
composite psi X 1O3
Modulus tension
composite psi X 10%
Ultimate strength
tension tangential
compact tables
psiX IO:’
No. of
specimens
Standard
deviation
Skull to skull
significant
differences
No. of
donors
Mean
181
14
0.272
0.047
Yes
179
14
0.108
oq42
Yes
240
14
0.051
0.019
Yes
237
26
3.5
2.1
Yes
65
11
0.53
0.4
No
219
14
8.1
4.4
Yes
122
14
0.19
0.08
No
327
18
0.22
o-11
No
237
26
10.7
5.1
Yes
210
14
14
5.2
No
237
26
97
80
No
210
14
51
32
No
237
26
1200
700
Yes
189
14
480
440
No
181
14
31.6
9.3
No
181
14
34.2
8.0
No
348
52
17
8
3.1
2.5
0.5
1.2
No
90
14
3.2
0.8
No
90
14
2.0
1,4
No
37
8
6.3
2.7
No
37
8
7.8
4.2
No
32
11
11.5
3.8
No
No
J. H. McELHANEY
502
et al.
Table 1.(cont.)
Property
No. of
specimens
No. of
donors
Mean
32
11
1.78
Mcdulus tension
compact tables
psi X IO5
Standard
deviation
Skull to skull
significant
differences
0.3
No
Table 2. Physical properties of Rhesus monkey (Macaca mufatta) cranial bone
Property
Thickness
in.
Dry weight
density #/ins
Ultimate strength
compression tangential
psi X 1oJ
Modulus
compression tangential
psi X lo5
Microhardness Vickers DPH
inner table
Microhardness Vickers DPH
outer tabIe
No. of
Specimens
Mean
Standard
deviation
Skull to skull
significant
differences
70
0.101
o-010
No
70
0.065
0.010
No
7
No
70
13.4
70
9-4
4-6
No
70
32.4
12.2
No
70
34
10.2
No
Figure
6 presents
typical stress-strain
curves for human skull bone in compression.
The modulus for tangential compression was
generally more than 2X larger than for radial
compression. Likewise, the ultimate strength
was generally higher in the tangential direction
than in the radial one. The ultimate strain and
energy absorption capacity, however, were
much larger for compression
in the radial
direction than the tangential direction.
Figure 7 shows three types of curves that
were observed for radial compression. The
upper curve is a classical compression curve
and occurred in approximately 50 per cent of
the tests. This type of curve indicates reasonably uniform stress with all parts of the specimen sharing the load. The middle curve shows
progressive failure indicating that some trabeculae through the diploe layer were more
highly loaded than others and failed or buckled
prematurely.
Since this type of curve was
observed in approximately 25 per cent of the
tests, the slope of the curve above the ‘knee’
is reported as a secondary modulus. The
lower curve represents a test where the diploe
was very porous and the trabeculae buckled
throughout. Although this is a true response
to compression load, the stress distribution
is quite non-homogeneous and not comparable
to the other types. For this reason, tests that
indicated significant buckling were culled
from the data and are not included in the tabulated results.
Figures 9 and 10 are maps of the diploE
thickness and modulus (radial) of the human
calvarium. These figures are three-dimensional displays drawn by computer and show
graphically the variation of a measured property over a surface. In this case, the surface
is developed from the projections of a grid on
the calvarium.
Figures 11 and 12 show the variation of
MECHANICAL
PROPERTIES OF CRANIAL
503
BONE
Fig. 6. Typical stress-strain curves for human skull bone in compression.
the compressive failure stress and modulus
(radial) over the surface of the skull. This
method of display contains much more detail
but lacks the graphical impact of the computer
plots. Fine grid maps of this type were examined in detail in an attempt to locate regions
of weakness. It was observed that in general
areas away from the sutures showed a much
thicker diploe with a corresponding reduced
c
0
At
radial compressive
modulus and ultimate
strength. However,
the energy absorption
capacity in these areas was higher. The
problem of regions of weakness is therefore a
complex and poorly defined one, requiring
much more work for its solution.
Correlations with density
Regression
analyses were
Classical
/’
I’
,’
25
Progressive
50
75
10
125
175
2@0
Fig. 7. Types of stress-strain curves for human skull bone in radial
compression.
225
performed
to
504
J. H. McELHANEY
er al.
TEllSlOllOUTER
MLE
I
ii
8
1
2
3
sdl”
Fig. 8. Typical stress-strain
correlate those properties
that were anticipated to be functionally related. The following relationships were found for human cranial
bone using data from those tests that produced classical stress-strain curves.
Radial compression
ER = (36y-
6
UC5
1
8
1
I
8
curves for human skull bone.
ER =
(2.02 X
cr = (2.9
x
1012y5.13)
psi; C, = O-86
IO+ ER)psi; C, = O-78
u = 1.2 x 108y3.3;c, = 0.91.
Tangential compression
1.3) psi
CDPWIAL
X 106;
C, = 0.62
Et= (46~ - l-52) psi
SWMRE
SaCI~IIIL tumt
Fig. 9. Computer drawn presentation of diploe layer thickness.
X
106; C, = O-65
MECHANICAL
PROPERTIES
OF CRANIAL
SAC#lfAl
Fig. IO. Computer
drawn presentation
u = (7707 - 26) psi X 103; C, = O-65
CT= 2.4
X
10e2 Et psi; C, = 0.57
where: ER is the modulus of elasticity loaded
in the radial direction (psi)
Et is the modulus of.elasticity loaded
in the tangential direction (psi)
u is the ultimate compressive strength
(psi)
y is the dry weight density (lb/in3)
C, is the correlation coefficient.
The values of the correlation coefficients
indicate that a significant part of the variation in the modulus and strength may be
attributed to variations in the density. The
relation between density and modulus and
density and strength was non-linear as evidenced by the higher values of the correlation coefficients or the logarithmic regression
analysis compared with the linear regression
analysis.
The relation between the stress and modulus
was approximately
linear however and indicated that the maximum strain may be
taken as a constant value independent
of
porosity. This provided a basis for a maximum
of compressive
BONE
505
SITRPC
modulus of elasticity.
strain theory of failure for skull bone with
failure strains of 2.9 X lo+? for radial compression and 2.4 X 10e2 for tangential compression indicated by the regression analysis.
It should be recognized, however, that these
regression analyses were performed only on
the data from the classical stress-strain curves,
and those specimens that failed progressively
or by buckling had much higher maximum
strain values. The regression equations given
are approximately
valid for dry weight densities between O-061 and O-042 #/in” and
should not be extrapolated beyond this range.
Porous block model
A physical - mathematical model has been
developed that allows the prediction of the
modulus and strength of bone based on its
density and internal geometry (Alem, 1969).
The accuracy of this prediction is limited by
the natural variation of the properties of compact bone, the base material of the model.
The advantage of such a model is that it allows
the description, in a relatively compact form,
of the physical properties of all types of bone,
ranging from the dense bone of the femoral
shaft to the very porous bone of the vertebral
body (Currey, 1964; Stech, 1966). A model of
506
J. H. McELHANEY
et al.
24
ts
14.
1
13 ‘4.5
%
Q
I
12, -4.7
a
tl
ii ,,. 6. I [
!z
ii
to -6. I C
I
9- 5.7(
0. 1 n
Fig. 11. Compressive failure stress.
this type may provide some insight into failure
patterns and hopefully into treatment of multiple myeloma, osteoporosis
and other diseases affecting the density of bone.
The primary assumption upon which this
model is based is that all bone has the same
physical properties in the small. That is, different types of bone have the same microscopic
properties but varying microscopic structure.
In dealing with the varying structure, it is
further assumed that local mechanical responses like the modulus of elasticity and
ultimate strength are proportional to the local
density raised to some power(n).
Consider the porous block model shown in
Fig. 13 which consists of a number of small
cubical aggregates arranged to form a larger
porous cube. In this model it is assumed that
MECHANICAL
PROPERTIES
OF CRANIAL
BONE
507
Fig. 12. Compressive modulus of elasticity.
the absence of one or more of the cubical
aggregates from the large block does not alter
either the position or the behavior of the
neighboring aggregates. The ratio of the void
volume to the total volume is called the
porosity C,
thus
C=
1 -rho
where y is the density of the sample and ‘y. is
the density of the base material.
A study of serial sections of cranial bone
samples, conducted as part of the research
program, has shown that the voids or marrow
spaces are approximately
homogeneously
distributed in directions tangent to the inner
and outer tables but in the radial direction
there is a continuous increase in porosity as
J. H. McELHANEY
508
et al.
form for Cch) the porosity of the tables is
c JlOO.
The average porosity is then
ra ial
lb
h
Cme-1842th-li2Pdh
taa
0
tangcatial(X)
Fig. 13. Poroui block model for cancellous bone.
one moves either from the inner or outer table
toward the center. Thus, the model indicates
that the modulus of elasticity of cranial bone
samples loaded tangentially may be simulated
by juxtaposition of a large number of slabs or
springs loaded in parallel while radial loading
is simulated by a large number of layers or
springs loaded in series. Therefore,
h
E
2?=
E,
I
for radial compression,
and
0
h
$=
j- [l0
0
C(h)]“dh
for tangential compression,
where E, is the modulus of the base material
and C(h) is the porosity of a slab or layer.
Additional study of serial sections of cranial
bone indicated that the variation of porosity
in the radial direction C(h) could be satisfactorily approximated by a Gaussian function
of the form
C,,, = Cm-1842'h-"2".
where C, is the maximum porosity. With this
It was found that for skewed distributions
the modulus of the composite is independent
of the location of C, and no loss in generality
is incurred by considering only symmetric
distributions. These integrals do not possess
closed form solutions but can be easily solved
numerically on a digital computer.
Various values of the exponent n were examined by comparing the correlation
coefficient relating the data to the model prediction. The value n = 3 was found to yield a best
fit with a correlation coefficient of O-79. In
addition, several other porosity distributions
were examined including parabolic and sinusoidal functions. The sinusoidal distribution
did not give significantly different results from
the Gaussian form presented here. Figures 14
and 15 show the relationship between modulus
and porosity predicted by the model and compared with the regression analyses previously
described.
The value of E,, the modulus of the base
material used to normalize the ordinate, was
1.8 X lo6 psi. This value was chosen to provide
a best fit with the data and agrees well with
the average value of 160 tension and compression tests of embalmed compact femoral
bone (1.84 X lo6 psi SD = O-41 X 10” psi) performed by the authors but as yet unreported,
and with the 32 tension tests of predominately
fresh compact bone from the inner and outer
tables of the cranium (1-78X 106 psi, SD =
O-3 x 106psi) reported here. The porosity was
based on a value of 0.068 #/in3 for the density
y. of compact bone which was the value
determined from the above mentioned femoral
and cranial specimens. The homogeneously
distributed porosity version of this model
agrees well with data for human vertebral
bodies tests with porosities ranging to 65 per
MECHANICAL
PROPERTIES
OF CRANIAL
BONE
509
Fig. 14. Normalized radial modulus vs. porosity, human cranial
bone in compression.
cent at one extreme and to data for femoral
compacta tests with a porosity of 0 per cent
(by definition) at the other.
The model indicates that with a Gaussian
distribution of the form used here the average
porosity cannot exceed 41 per cent because
the maximum porosity at the center of the
diplok! can not be greater than 100 per cent.
The value of the modulus is therefore much
more sensitive to porosity distribution in the
high porosity material than in the low porosity
material.
Figure 16 shows the model’s prediction of
the ratio of the radial modulus to the tangential modulus compared with measured values.
The model explains quite well the observed
increase of non-isotropic response with increasing porosity. The non-isotropic mechanical response of the model is based on the
non-isotropic porosity distribution and is very
sensitive to the form of the porosity distribution in the region approaching 41 per cent
average porosity. It is probable that much of
the lack of fit in the low density region may be
Fig. 15. Normalized tangential modulus vs. porosity, human cranial
bone in compression.
510
J. H. McELHANEY
et al.
is shown by comparing the Gaussian model
prediction with the homogeneous model prediction. The value of the exponent used to
obtain these model responses was 4 and the
correlation coefficient comparing the model
and the data was 0.67. In the low porosity
range the Gaussian distribution yielded a
satisfactory fit while in the high porosity range
the homogeneous model gave better results.
DISCUSSION
Fig. 16. Ratio of radial to tangential moduli vs. porosity.
explained by the difference between the actual
porosity distribution in the sample and the
assumed Gaussian distribution.
Figure 17 shows the response of the porous
block model developed from the strength data.
The ultimate strength of the base material
was taken as 21,000 psi to obtain a best fit.
This corresponds approximately to the long
bone data, Evans (1957) and our own baseline experiments on compact femoral bone in
compression (19,400 psi, SD = 6800 psi). In
this figure the effect of porosity distribution
1)
15
30
lrerrgr
Many of the mechanical responses of cranial
bone are strongly influenced by the structural
arrangement of the dipl& (Endo, 1966, Gurdjian et al., 1947). Thus in these tests properties such as compressive strength and modulus
are structural properties and the large values
of the standard deviations observed for these
properties are primarily due to variations in
the porosity and internal arrangement of the
traebeculae. The microhardness tests and the
tension test of the inner and outer tables indicate that a single material porous block
model is justified as a first approximation in
describing the relationship between structure
and mechanical response. Relating the value
of a property to the density raised to some
power n and then determining n empirically
provides a means of incorporating in the
model many of the structural elements that
influence the response but are too complex to
25
30
35
4.
45
Pwrsit! %
Fig. 17. Strength vs. porosity, human cranial bone in compression.
MECHANICAL
PROPERTIES
be included in detail. The model presented
here is therefore complex enough to explain
much of the property variation observed but
much too simple to explain the mechanisms
involved. The model shows that the modulus
of bone is approximately proportional to the
third power of the density and the strength
is proportional
to the fourth power of the
density. Thus, small porosity changes in bone
of low relative density result in only small
changes in strength and modulus while small
porosity changes in bone of high relative
density result in large changes in strength and
modulus. The porosity distribution in a given
sample of bone is much more significant in its
effect on strength and modulus in bone of low
relative density than in bone of high relative
density. Of interest is the fact that the homogeneous version of this model fits much of
Coble’s and Kingery (1956) data on various
porous ceramics indicating that bone is not
unique in its response to porosity variations.
The material properties in the small, i.e.
hardness,
density and local compression
strength of human and primate (Macaca
mufatta) cranial bone, are not significantly
different. The amount and distribution of the
diploe, however, is quite variable and therefore the structural responses-in
particular
the energy absorption, gross stiffness and
damping characteristics
which are strongly
dependent
on structure-will
vary greatly.
Acknon~ledgements-The
results reported in this paper
represent a composite compilation of work performed in
the Biomaterials Laboratory, H.S.R.I., The University
of Michigan and the Biomechanics Laboratory, T.A.M.,
West Virginia University. Particular acknowledgement
is given to the contributions of Jack Wood, Iqbal Barodawala, and Hurley Robbins from The University of Michigan, and Bruce~Ours, Frank Ammons, George Utt and
Richard Stahraker from West Virginia Universitv who
performed much of the work presented here.
_
The research upon which this publication is based was
performed pursuant to Contract No. PH43-67-1137 with
the National Institutes of Health, National Institute of
OF CRANIAL
BONE
511
Neurological Diseases and Stroke, Public Health Service,
Department of Health, Education and Welfare.
REFERENCES
Alem, N. M. (1969) A model for skull bone in compression. Master’s Thesis, West Virginia University,
Morgantown, West Virginia.
Coble, R. L. and Kingery, W. D. (1956) Effect of porosity
on physical properties of sintered aluminea. J. Am.
ceramic Sot. 29,377.
Currey, J. D. (I 964) Three analogies to explain the mechanical properties of bone. Biorheology 2. 1- 10.
Dempster, W. T. (1967) Correlation of types of cortical
grain structure with architectural features of the human
skull. Am. J. Anat. lZO, l-32.
Endo, B. (1966) A biomechanical study of the human
facial skeleton by means of strain sensitive lacquer.
Okajimas Pol. A nat. Jap. 42,205-2 17.
Evans, F. G. (I 957) Stress and Strain in Bones. Thomas,
Springfield, Illinois.
Evans, F. G. and Lissner, H. R. (1957) Tensile and compressive strength of human parietal bone. /. appl. Physiol. l&492-497.
Evans, F. G. and Bang, S. (1967) Differences and relationships between the physical properties and the
microscopic structure of human femoral, tibia1 and
fibular cortical bone. Am. J. Anat. lu), 79-88.
Gurdjian, E. S. and Lissner, H. R. (1947) Deformations
of the skull in head injury as studied by the ‘stresscoat’
technique. Am. J. Surg. 73,269-28 1.
Gurdjian, E. S., Lissner, H. R. and Webster, J. E. (1947)
The mechanism of production of linear skull fracture.
Surgery Gynec. Obstet. 85, 195-210.
McElhaney, J. H., Fogle, J., Byars, E. and Weaver, G.
(1964) Effect of embalming on the mechanical properties of beef bone. 1. appl. Physiot. 19-6, l234- 1236.
McElhaney, J. H. and Byars, E. F. (1966) Dynamic
response of bone and muscle tissue. J. appl. Physiol.
21.4.
Melvin, J. W., Robbins, D. H. and Roberts, V. L. (1969)
The mechanical behavior of the diplof layer of the
human skull in compression. Dee. Mech. 5,s 1 l-8 18.
Stech, E. L. (1966) A descriptive model of lamellar bone
anisotronv. ASME Pager No. 66-HUF-3.
Wood, J. ’ L. (1969) Mechanical properties of human
cranial bone in tension. Ph.D. Dissertation, The University of Michigan, Ann Arbor, Michigan.
Zeniya, T., Takazono, K. and Ibuki, S. (1964) On the
Vickers hardness of human temporal bone. J. Kyoto
Pref. Med. Univ. 73, 309-310 (Japanese text with
English summary).
A Survey of Current Head injury Research. A report
prepared by the subcommittee on Head Injury, National
Advisory Neurological Diseases and Stroke Council,
National Institutes of Health, Bethesda, Maryland,
1969.